We've seen every piece on the board now, except the knight. So lets set up the initial position (like we learned way back when) and look at it.
Ignoring the knights for the moment, lets consider this position and how the pieces move. We know from previous examples that the pawns, kings, queens, bishops, and rooks can move (in their own fashion) to any empty square in their path. But we also know none of them can jump over anything that's in their way. So white and black both have whole rows of pawns in the way that are going to have to move before any bishops, queens, rooks, or kings are going to be able to move anywhere. But do knights have this problem? Do they have to wait? No, they don't; knight are unique in that they can jump over other pieces, either of their own color or those of your opponent's! 
Now there's two ways to look at the way a knight moves. One or the other will probably work for you just fine.
The first way to look at the knight move might be as an 'L'shaped move.

Regarding the knight move as 'L'shaped is
one way of looking at it, and its how a lot of people learned.
Lets just take a knight and drop it in the middle of an empty board on square 'e4'. The red dots show all of the squares to which this knight can move from 'e4'. Notice that they're all darkcolored squares. That's because the knight sits on a lightcolored square. In this example we're going to move the knight from 'e4' to the 'f6' square (you would write that down as 'Nf6'). 
To get an idea of how the knight would move, we'll take a red marker and draw the path on the board that the knight will take to get to the red dot on square 'f6', and we'll end up with this:
The path the knight is going to take looks
here like an upside down 'L'.
Now lets work it out in detail in three steps: 

Step 1: the knight moves forward one square. Its a third of the way there. 

Step 2: The knight moves forward another square. We're getting there. One last thing to do. 

Step 3: The knight reaches its destination on square 'f6', and its move is now complete. 
Now of course, knights aren't limited to moving forward like pawns. So let's toss a few more knights on the board and draw some (but not all) of the 'L'shaped moves for them.
To try to better tell the difference, the
black knights' moves are drawn with a blue marker.
Each of these knights has more than just the one drawn move available to it. The red dots show all of the available moves. Can you tell all of the squares to which the knights as placed here can move? (The answer is further down the page.) 
Another way to consider the knight move might be as a combination of a onesquare straight move, followed by a onesquare diagonal move, which I call the 'straightthendiagonal' method.

This method might be easier for some people to figure out, since there's only two steps to follow. Once again, we drop a knight in the middle of the board on square 'e4' and we're going to move the 'f6' square. 
Now lets get out our red marker and show the available moves using the straightthendiagonal method.
As you can see here, the knight takes a straight move along the row (rank) or column (file) for one square, then makes a one square diagonal move to arrive at its final destination. Straight, then Diagonal. 
To break it out into a two step operation, going from 'e4' to f6', we do the following:

Step 1: the knight moves forward straight along the file, just as if it was making a onesquare pawn move. But its not done yet... 

Step 2: the knight makes a onesquare diagonal move to arrive at its final desination on square 'f6'. 
Now, if you're still having a tough time visualizing all the squares that a knight can move to on the board, don't worry about it; let's see if we can help.

We'll take a knight and dump it in the middle of the board again on 'e4'. 
"Well yeah, Mr. Chessguy, but where all can the knight move?" Well, let's add some target dots. Remember, now, that a knight can only move to an oppositecolored square from the one on which its sitting. That is, THIS knight is sitting on a white square, so it can only move to darkcolored squares.

See? All the dots are on dark squares. Any better? "Well, I suppose so, Mr. Chessguy, but all those dots look like a lot to remember, as well." 
I agree, but it looks to me like the dots might form a circle. Lets see....
Is that any clearer?
This is what I refer to as the knight's "circle of influence". This one circle shows all of the darkcolored squares to which this lightsquared knight can move. Every darkcolored square that the circle goes through is a square that can be occupied by knight simply by moving it from 'e4'. 
Now lets look at a morecomplicated position, with a lot of knights. Don't worry, this won't be tough.

Can you see all the squares to which the knights might move? Let's add a little help. 

We've seen this position before, already.
"Well, okay, what I see now is a whole lot of dots, and some of them are right next to each other. Which knights can go to which dots?" 
All right, lets draw the knights' "circles of influence" on the board.
That ought to help. (The white knights have
red circles, the black knights have blue circles.)
The white knight on lightcolored square 'e4' can move to any of dark squares 'c5', 'd6', 'f6', 'g5', 'g3', 'f2', 'd2', and 'c3', as shown by the complete red circle. Dark squares 'd2' and 'c3' also come under the circle of influence from the white knight on square 'b1'. The 'b1' square knight's partial circle covers the already mentioned 'd2' and 'c3', and also 'a3'. The circle of influence for the black knight on 'g8' (towards the top righthand corner of the board consists of squares 'e7', 'f6', and 'h6'. (You might have already noticed that 'f6' is also covered by the white knight on 'e4'.) And remember that knights can only move to their oppositecolored squares? Well, the knight sitting on dark square 'b8' (upper left) can only move to lightcolored squares 'a6', 'c6', and 'd7'. 
That pretty much covers knight moves, but I want to point out something else, related to the "circles of influence" position above. Notice that the two black knights each cover 3 squares on the board, and each one has only a partial circle. The white knight on 'b1' also only covers 3 squares and has a circle fragment. The other white knight on square 'e4', however, covers 8 squares on the board, and enjoys a full and complete circle of influence. This is because the knight is at the center of the board, rather than at the edge. This illustrates a general rule of thumb that I want you to try to remember as we go along, and that is knights that are closer to the center of the board will generally be more valuable to you than knights that are stuck along the edges.